What Is The Center Of A Circle Represented By The Equation { (x+9) 2+(y-6) 2=10^2$}$?A. { (-9, 6)$}$ B. { (-6, 9)$}$ C. { (6, -9)$}$ D. { (9, -6)$}$

Understanding the Equation of a Circle

The equation of a circle in standard form is given by ${(x-h)^2+(y-k)^2=r^2}$, where ${(h,k)}$ represents the coordinates of the center of the circle, and ${r}$ is the radius. In the given equation ${(x+9)^2+(y-6)^2=10^2}$, we can identify the values of ${h}$, ${k}$, and ${r}$.

Identifying the Center of the Circle

To find the center of the circle, we need to identify the values of ${h}$ and ${k}$ in the equation. In the given equation, the value of ${h}$ is ${-9}$, and the value of ${k}$ is ${6}$. Therefore, the center of the circle is represented by the coordinates ${(-9, 6)}$.

Analyzing the Options

Let's analyze the options given in the problem:

• A. ${(-9, 6)}$
• B. ${(-6, 9)}$
• C. ${(6, -9)}$
• D. ${(9, -6)}$

Conclusion

Based on the analysis, the correct answer is option A. ${(-9, 6)}$. This is because the value of ${h}$ in the equation is ${-9}$, and the value of ${k}$ is ${6}$, which corresponds to the coordinates ${(-9, 6)}$.

Final Answer

The final answer is ${(-9, 6)}$.

Understanding the Equation of a Circle

The equation of a circle in standard form is given by ${(x-h)^2+(y-k)^2=r^2}$, where ${(h,k)}$ represents the coordinates of the center of the circle, and ${r}$ is the radius. In the given equation ${(x+9)^2+(y-6)^2=10^2}$, we can identify the values of ${h}$, ${k}$, and ${r}$.

Identifying the Center of the Circle

To find the center of the circle, we need to identify the values of ${h}$ and ${k}$ in the equation. In the given equation, the value of ${h}$ is ${-9}$, and the value of ${k}$ is ${6}$. Therefore, the center of the circle is represented by the coordinates ${(-9, 6)}$.

Analyzing the Options

Let's analyze the options given in the problem:

• A. ${(-9, 6)}$
• B. ${(-6, 9)}$
• C. ${(6, -9)}$
• D. ${(9, -6)}$

Conclusion

Based on the analysis, the correct answer is option A. ${(-9, 6)}$. This is because the value of ${h}$ in the equation is ${-9}$, and the value of ${k}$ is ${6}$, which corresponds to the coordinates ${(-9, 6)}$.

Final Answer

The final answer is ${(-9, 6)}$.

Q&A: Understanding the Center of a Circle

Q: What is the standard form of the equation of a circle?

A: The standard form of the equation of a circle is given by ${(x-h)^2+(y-k)^2=r^2}$, where ${(h,k)}$ represents the coordinates of the center of the circle, and ${r}$ is the radius.

Q: How do I identify the center of a circle from its equation?

A: To find the center of the circle, you need to identify the values of ${h}$ and ${k}$ in the equation. In the given equation ${(x+9)^2+(y-6)^2=10^2}$, the value of ${h}$ is ${-9}$, and the value of ${k}$ is ${6}$.

Q: What is the significance of the values of ${h}$ and ${k}$ in the equation of a circle?

A: The values of ${h}$ and ${k}$ in the equation of a circle represent the coordinates of the center of the circle.

Q: How do I determine the correct answer from the given options?

A: To determine the correct answer, you need to analyze the options and compare them with the values of ${h}$ and ${k}$ in the equation. In this case, the correct answer is option A. ${(-9, 6)}$.

Q: What is the final answer to the problem?

A: The final answer is ${(-9, 6)}$.

Q: What is the radius of the circle represented by the equation ${(x+9)^2+(y-6)^2=10^2}$?

A: The radius of the circle is ${10}$, since the equation is in the form ${(x-h)^2+(y-k)^2=r^2}$.

Q: How do I find the radius of a circle from its equation?

A: To find the radius of a circle from its equation, you need to identify the value of ${r}$ in the equation. In this case, the value of ${r}$ is ${10}$.

Q: What is the significance of the radius of a circle?

A: The radius of a circle represents the distance from the center of the circle to any point on the circle.

Q: How do I determine the correct answer from the given options?

A: To determine the correct answer, you need to analyze the options and compare them with the values of ${h}$, ${k}$, and ${r}$ in the equation. In this case, the correct answer is option A. ${(-9, 6)}$.